Chapter 11: Further Topics in Algebra 11.1 Sequences and Series 11.2 Arithmetic Sequences and Series 11.3 Geometric Sequences and Series 11.4 Counting Theory 11.5 The Binomial Theorem 11.6 Mathematical Induction 11.7 Probability

11.4 Counting Theory Example If there are 3 roads from Albany to Baker and 2 Roads from Baker to Creswich, how many ways are there to travel from Albany to Creswich? Solution The tree diagram shows all possible routes. For each of the 3 roads to Albany, there are 2 roads to Creswich. There are 3.2=6 different routes from Albany to Creswich.

11.4 Counting Theory Two events are independent if neither influences the outcome of the other. For example, choice of road to Baker does not influence the choice of road to Creswich. Sequences of independent events can be counted using the Fundamental Principle of Counting.

11.4 Counting Theory Fundamental Principle of Counting If n independent events occur, with m1 ways for event 1 to occur, m2 ways for event 2 to occur, … and mn ways for event n to occur, then there are different ways for all n events to occur.

11.4 Using the Fundamental Principal of Counting Example A restaurant offers a choice of 3 salads, 5 main dishes, and 2 desserts. Count the number of 3-course meals that can be selected. Solution The first event can occur in 3 ways, the second event can occur in 5 ways, and the third event in 2 ways; thus there are possible meals.

11.4 n-Factorial n-Factorial For any positive integer n, By definition Example Evaluate (a) 5! (b) 7! Solution (a) (b)

11.4 Permutations A permutation of n elements taken r at a time is one of the arrangements of r elements from a set of n elements. The number of permutations of n elements taken r at a time is denoted P(n,r). Example An ordering of 3 books selected from 5 books is a permutation. There are 5 ways to select the first book, 4 ways to select the second book, and 3 ways to select the third book. Thus, there are permutations.

11.4 Permutations Permutations of n Elements Taken r at a Time If P(n,r) denotes the number of permutations of n elements taken r at a time, with r < n, then

11.4 Using the Permutation Formula Example Suppose 8 people enter an event in a swim meet. In how many ways could the gold, silver, and bronze prizes be awarded. Solution ways

11.4 Combinations A subset of items selected without regard to order is called a combination. The number of combinations of n elements taken r at a time is denoted C(n,r) or nCr and is sometimes read as “n choose r”.

11.4 Combinations Combinations of n Elements Taken r at a Time If C(n,r) represents the number of combinations of n things taken r at a time, with r < n, then

11.4 Using the Combinations Formula Example How many different committees of 3 people can be chosen from a group of 8 people? Solution Since a committee is an unordered set (the arrangement of the 3 people does not matter,) the number of committees is

11.4 Distinguishing between Permutations and Combinations Different orderings or arrangements of the r objects are different permutations. Clue words: arrangement, schedule, order Each choice or subset of r objects gives one combi-nation. Order within the group of r objects does not matter. Clue words: group, committee, sample, selection